Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle BOC = 2x + 20$, and $ m \angle AOB = 8x - 60$, find $m\angle BOC$. $O$ $A$ $C$ $B$
Answer: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {8x - 60} + {2x + 20} = {90}$ Combine like terms: $ 10x - 40 = 90$ Add $40$ to both sides: $ 10x = 130$ Divide both sides by $10$ to find $x$ $ x = 13$ Substitute $13$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 2({13}) + 20$ Simplify: $ {m\angle BOC = 26 + 20}$ So ${m\angle BOC = 46}$.